Systems of linear differential equations

The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example. Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, C,. and C, for example, can be eliminated from the equations for and which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 

In this section, we will outline only those properties of the Laplace transform that are directly relevant to the solution of systems of linear differential equations with constant coefficients. A more extensive coverage can be found, for example, in the text book by Franklin [6]. 

To conclude this section on two-compartment models we note that the hybrid constants a and p in the exponential function are eigenvalues of the matrix of coefficients of the system of linear differential equations ... 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

R.I. Jennrich and P.B. Right, Fitting systems of linear differential equations using computer generated exact derivatives, Technometrics,... 

Equation (4.2.7) is a homogeneous system of linear differential equations. The search for a particular solution in the form... 

Equation 1.12 is a system of linear differential equations with constant coefficients. Then, following the rules for solving this type of an equation, its solution can be written in the following form [7] ... 

Therefore, Fick s law, when applied to all elements of the compartmental structure, leads to a system of linear differential equations. There are as many equations as compartments in the configuration. If we set... 

Figure 3.4 Stability types of particular points in the system of linear differential equations dy/dt = ay + bz dz/dt = cy + dz in coordinates (A, y) (according to A. M. Lyapunov).

Laplace transforms can be used to transform this system of linear differential equations in the time domain into a system of linear equations in the Laplace domain. From the table of Laplace operations (Appendix I) we obtain... 

Bates, D. M., and D, G. Watts, Multiresponse estimation with special application to systems of linear differential equations, Technometrics, 27, 329-339 (1985). 

Linearity. A system is linear when its response to a sum of individual input signals is equal to the sum of the individual responses. This also implies that the system is described by a system of linear differential equations [see e.g., Eqs. (2) and (7)]. Electrochemical systems are usually highly nonlinear and the impedance is obtained by the linearization of equations [see e.g., Eqs. (42) and (130)] for small amplitudes. For linear systems, the response is independent of the amplitude. It is easy to verify the linearity of the system if the impedance obtained is the same when the amplitude of the applied ac signal is halved, then the system is... 

The solution of any other linear differential equation or of a system of linear differential equations will follow the same general pattern outlined in the two examples. 

What is the complementary solution, and what is the particular solution for (a) an nth-order linear differential equation, and (b) a 2 x 2 system of linear differential equations What do these solutions mean What factors determine them ... 

Eq. (5.10) represents a system of linear differential equations with constant coefficients, if the quantum yield does not depend on the concentrations. Any photoreaction, which can be rewritten by the transformation eq. (5.9) to such a system, is called a quasi-linear photoreaction in the following. All the relationships derived in Section 2.2 can be used, if the real reaction time is substituted by the transformed time 0. 

Systems of linear differential equations for which the stiffiiess is greater than 1000 or those for which the solution involves periodic functions present the greatest problems for the CBCCDS integrator. 

On periodic solutions of systems of linear differential equations with lag. Dokl. Akad. Nauk Ukrain.SSR, Ser. A, 1970(3), 217-220 (Ukrainian). 

On reducibility of systems of linear differential equations with quasiperiodic coefficients. Ukrain. Mat. Zhum., 20, (1968), 279-281. 

Controllability of Linear Systems It is possible to determine if a system of linear differential equations is controllable or not. Although reactive systems found in AR theory are generally nonlinear, the underlying concepts are similar and shall be useful for later discussions. In 1959, Rudolf Kalman showed that specifically for a linear, time-invariant system, it is possible to determine whether a system is controllable by computing the rank of a special controllability block matrix, E (Kalman, 1959)... 

Using the fact that C q) = 0 and keeping only the dominant terms for y 1, we are left with a system of linear differential equations... 

A calorimeter is usually regarded as an object that can be described by one differential equation or a system of linear differential equations with constant coefficients. These equations are treated as the mathematical models of calorimeters. If there are many output functions, then the dynamic object (calorimeter) is described by a system of n differential equations. Assuming linearity and applying the superposition rule, one... 

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